So my father showed my relatives and me this trick over the holiday. Nobody was able to work it out and in fact I didn’t figure it out until I worked it out algebraically on Friday. I explained it to my dad then and he got it, although he did say he wouldn’t tell anybody else the secret.
If you write out the instructions you find that the final number is of the form 9x, and all the 9 multiples in the list, except 90 which won’t be the final number anyway, have the same symbol everytime. The symbols change but the 9 multiples are always the same.